
Matheology § 074
Posted:
Jul 13, 2012 2:29 PM


Matheology § 074
Dedekind tried to describe an infinite class by saying that it is a class which is similar to a proper subclass of itself. [...] I am to investigate in a particular case whether a class is finite or not, whether a certain row of trees, say, is finite or infinite. So, in accordance with the definition, I take a subclass of the row of trees and investigate whether it is similar (i.e. can be coordinated oneto one) to the whole class! (Here already the whole thing has become laughable.) It hasn?t any meaning; for, if I take a "finite class" as a subclass, the attempt to coordinate it with the whole class must eo ipso fail: and if I make the attempt with an infinite class ? but already that is a piece of nonsense, for if it is infinite, I cannot make an attempt to coordinate it. [...] An infinite class is not a class which contains more members than a finite one, in the ordinary sense of the word "more". If we say that an infinite number is greater than a finite one, that doesn't make the two comparable, because in that statement the word "greater" hasn?t the same meaning as it has say in the proposition 5 > 4!
The form of expression "m = 2n correlates a class with one of its proper subclasses" uses a misleading analogy to clothe a trivial sense in a paradoxical form. (And instead of being ashamed of this paradoxical form as something ridiculous, people plume themselves on a victory over all prejudices of the understanding). It is exactly as if one changed the rules of chess and said it had been shown that chess could also be played quite differently.
When "all apples" are spoken of, it isn?t, so to speak, any concern of logic how many apples there are. With numbers it is different; logic is responsible for each and every one of them.
Mathematics consists entirely of calculations.
In mathematics description and object are equivalent. {{Therefore numbers that cannot be described cannot exist.}} "The fifth number of the number series has these properties" says the same as "5 has these properties". The properties of a house do not follow from its position in a row of houses; but the properties of a number are the properties of a position.
[L. Wittgenstein: "Philosophical Grammar", Basil Blackwell, Oxford (1969), quoted from E.D.Buckner: "THE LOGIC MUSEUM" (2005), unfortunately no longer on the web.] http://www.amazon.de/PhilosophicalGrammarLudwigWittgenstein/dp/0631123504/ref=sr_1_1?ie=UTF8&qid=1286647502&sr=11
There is no path to infinity, not even an endless one. [§ 123]
It isn't just impossible "for us men" to run through the natural numbers one by one; it's impossible, it means nothing. [?] you can?t talk about all numbers, because there's no such thing as all numbers. [§ 124]
An "infinitely complicated law" means no law at all. [§ 125]
There's no such thing as "all numbers" simply because there are infinitely many. [§ 126]
Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes.
In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar. [§ 141]
Generality in mathematics is a direction, an arrow pointing along the series generated by an operation. And you can even say that the arrow points to infinity; but does that mean that there is something ? infinity ? at which it points, as at a thing? Construed in that way, it must of course lead to endless nonsense. [§ 142]
If I were to say "If we were acquainted with an infinite extension, then it would be all right to talk of an actual infinite", that would really be like saying, "If there were a sense of abracadabra then it would be all right to talk about abracadabraic sense perception". [§ 144] Set theory is wrong because it apparently presupposes a symbolism which doesn't exist instead of one that does exist (is alone possible). It builds on a fictitious symbolism, therefore on nonsense. [§ 174]
[L. Wittgenstein: "Philosophical Remarks", quoted from E.D.Buckner: "THE LOGIC MUSEUM" (2005), unfortunately no longer on the web]
Regards, WM

